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| -rw-r--r-- | paper/sections/lowerbound.tex | 92 | ||||
| -rw-r--r-- | paper/sections/results.tex | 11 |
2 files changed, 57 insertions, 46 deletions
diff --git a/paper/sections/lowerbound.tex b/paper/sections/lowerbound.tex index b714766..7f51f46 100644 --- a/paper/sections/lowerbound.tex +++ b/paper/sections/lowerbound.tex @@ -1,63 +1,61 @@ -In this section, we will consider the stable sparse recovery setting of -Section~\ref{sec:relaxing_sparsity}. Our goal is to obtain an information-theoretic -lower bound on the number of measurements necessary to approximately recover -the parameter $\theta$ of a cascade model from observed cascades. Similar lower -bounds were obtained for sparse linear inverse problems in \cite{pw11, pw12, -bipw11} +In \cite{Netrapalli:2012}, the authors explicitate a lower bound of +$\Omega(s\log\frac{n}{s})$ on the number of cascades necessary to achieve good +support recovery with constant probability under a \emph{correlation decay} +assumption. In this section, we will consider the stable sparse recovery +setting of Section~\ref{sec:relaxing_sparsity}. Our goal is to obtain an +information-theoretic lower bound on the number of measurements necessary to +approximately recover the parameter $\theta$ of a cascade model from observed +cascades. Similar lower bounds were obtained for sparse \emph{linear} inverse +problems in \cite{pw11, pw12, bipw11}. \begin{theorem} \label{thm:lb} Let us consider a cascade model of the form \eqref{eq:glm} and a recovery algorithm $\mathcal{A}$ which takes as input $n$ random cascade measurements and - outputs $\hat{\theta}$ such that with probability $\delta<\frac{1}{2}$ + outputs $\hat{\theta}$ such that with probability $\delta>\frac{1}{2}$ (over the measurements): - \begin{displaymath} + \begin{equation} + \label{eq:lb} \|\hat{\theta}-\theta^*\|_2\leq C\min_{\|\theta\|_0\leq s}\|\theta-\theta^*\|_2 - \end{displaymath} - where $\theta^*$ is the true paramter of the cascade model. Then $n + \end{equation} + where $\theta^*$ is the true parameter of the cascade model. Then $n = \Omega(s\log\frac{m}{s}/\log C)$. \end{theorem} This theorem should be contrasted with Theorem~\ref{thm:approx_sparse}: up to an additive $s\log s$ factor, the number of measurements required by our algorithm is -tight. +tight. The proof of Theorem~\ref{thm:lb} follows an approach similar to \cite{pw12}. +We only present a sketch of the proof here and refer the reader to their paper +for more details. -The proof of Theorem~\ref{thm:lb} follows an approach similar to \cite{pw12}. -Let us consider an algorithm $\mathcal{A}$ as in the theorem. Intuitively, -$\mathcal{A}$ allows Alice and Bob to send $\Omega(s\log\frac{m}{s})$ quantity -of information over a Gaussian channel. By the Shannon-Hartley theorem, this -quantity of information is $O(n\log C)$. These two bounds together give the -theorem. +Let us consider an algorithm $\mathcal{A}$ which verifies the recovery +guarantee of Theorem~\ref{thm:lb}: there exists a probability distribution over +measurements such that for all vectors $\theta^*$, \eqref{eq:lb} holds w.p. +$\delta$. This implies by the probabilistic method that for all distribution +$D$ over vectors $\theta$, there exists an $n\times m$ measurement matrix $X_D$ with +such that \eqref{eq:lb} holds w.p. $\delta$ ($\theta$ is now the +random variable). -Formally, let $\mathcal{F}\subset \{S\subset [1..m]\,|\, |S|=s\}$ be a family -of $s$-sparse supports such that: -\begin{itemize} - \item $|S\Delta S'|\geq s$ for $S\neq S'\in\mathcal{F}$ . - \item $\P_{S\in\mathcal{F}}[i\in S]=\frac{s}{m}$ for all $i\in [1..m]$ and - when $S$ is chosen uniformly at random in $\mathcal{F}$ - \item $\log|\mathcal{F}| = \Omega(s\log\frac{m}{s})$ -\end{itemize} -such a family can be obtained by considering a linear code (see details in -\cite{pw11}. Let $X = \big\{t\in\{-1,0,1\}^m\,|\, \mathrm{supp}(t)\in\mathcal{F}\big\}$. +Consider the following distribution $D$: choose $S$ +uniformly at random from a ``well-chosen'' set of $s$-sparse supports +$\mathcal{F}$ and $t$ uniformly at random from $X +\defeq\big\{t\in\{-1,0,1\}^m\,|\, \mathrm{supp}(t)\in\mathcal{F}\big\}$. Define +$\theta = t + w$ where $w\sim\mathcal{N}(0, \alpha\frac{s}{m}I_m})$ and $\alpha += \Omega(\frac{1}{C})$. -Consider the following communication game between Alice and Bob. +Consider the following communication game between Alice and Bob: \begin{itemize} - \item Alice chooses $S$ uniformly at random from $\mathcal{F}$ and - $t$ uniformly at random from $X$ subject to $\mathrm{supp}(x) = S$ - \item Let $w\sim\mathcal{N}(0, \alpha \frac{s}{m}I_m)$ and $\theta - = t + w$. Since for all $\theta$, $\mathcal{A}$ recovers $\theta$ with - probability $1-\delta$ over the measurements, there exists a fixed set - of measurements $x_1,\ldots, x_n$ such that with probability - $1-\delta$ over $\theta$, $\mathcal{A}$ recovers $\theta$. Alice sends - those measurements to Bob. - \item Bob uses $\mathcal{A}$ to recover $\hat{\theta}$ from the - measurements, then computes $\hat{t}$ the best $\ell_2$ approximation - of $\hat{\theta}$ in $X$. + \item Alice sends $y\in\reals^m$ drawn from a Bernouilli distribution of parameter + $f(X_D\theta)$ to Bob. + \item Bob uses $\mathcal{A}$ to recover $\hat{\theta}$ from $y$. \end{itemize} +It can be shown that at the end of the game Bob now has a quantity of +information $\Omega(s\log \frac{m}{s})$ about $S$. By the Shannon-Hartley +theorem, this information is also +upper-bounded by $\O(n\log C)$. These two bounds together imply the theorem. -We have the following from \cite{pw11} and \cite{pw12} respectively. - +\begin{comment} \begin{lemma} \label{lemma:upperinf} Let $S'=\mathrm{supp}(\hat{t})$. If $\alpha = \Omega(\frac{1}{C})$, $I(S, S') @@ -71,3 +69,15 @@ We have the following from \cite{pw11} and \cite{pw12} respectively. Lemmas \ref{lemma:lowerinf} and \ref{lemma:upperinf} together give Theorem \ref{thm:lb}. + +Formally, let $\mathcal{F}\subset \{S\subset [1..m]\,|\, |S|=s\}$ be a family +of $s$-sparse supports such that: +\begin{itemize} + \item $|S\Delta S'|\geq s$ for $S\neq S'\in\mathcal{F}$ . + \item $\P_{S\in\mathcal{F}}[i\in S]=\frac{s}{m}$ for all $i\in [1..m]$ and + when $S$ is chosen uniformly at random in $\mathcal{F}$ + \item $\log|\mathcal{F}| = \Omega(s\log\frac{m}{s})$ +\end{itemize} +such a family can be obtained by considering a linear code (see details in +\cite{pw11}. Let . +\end{comment} diff --git a/paper/sections/results.tex b/paper/sections/results.tex index e5aa861..ef205f4 100644 --- a/paper/sections/results.tex +++ b/paper/sections/results.tex @@ -347,16 +347,18 @@ Following similar notation, let $Q^* \defeq \nabla^2 f(\theta^*)$. Let $S$ and $ The (S,s)-irrepresentability holds if $\nu_{\text{irrepresentable}}(S) < 1 - \epsilon$ for $\epsilon > 0$ \end{definition} -It is intuitive that the irrepresentability condition is stronger than the -{\bf(RE)} assumption. In fact, a slightly modified result from -\cite{vandegeer:2009} shows that a `strong' irrepresentability condition -directly {\it implies} the {\bf(RE)} condition for $\ell_2$-recovery: +It is possible to construct examples where the (RE) condition holds but not the +irrepresentability condition \cite{vandegeer:2009}. In fact, a slightly +modified result from \cite{vandegeer:2009} shows that a `strong' +irrepresentability condition directly {\it implies} the {\bf(RE)} condition for +$\ell_2$-recovery: \begin{proposition} \label{prop:irrepresentability} If the irrepresentability condition holds with $\epsilon > \frac{2}{3}$, then the restricted eigenvalue condition holds with constant $\gamma_n \geq \frac{ (1 - 3(1 -\epsilon))^2 \lambda_{\min}^2}{4s}n$, where $\lambda_{\min} > 0$ is the smallest eigenvalue of $Q^*_{S,S}$, on which the results of \cite{Daneshmand:2014} also depend. \end{proposition} +\begin{comment} Furthermore, recent papers \cite{vandegeer:2011}, \cite{Zou:2006}, argue that the irrepresentability condition is unrealistic in situations where there is a correlation between variables. Consider the following simplified example from @@ -373,7 +375,6 @@ I_s & \rho J \\ where $I_s$ is the $s \times s$ identity matrix, $J$ is the all-ones matrix and $\rho \in \mathbb{R}^+$. It is easy to see that $\nu_{\text{irrepresentable}}(S) = \rho s$ and $\lambda_{\min}(Q) \geq 1 - \rho$, such that for any $\rho > \frac{1}{s}$ and $\rho < 1$, the restricted eigenvalue holds trivially but the (S,s)-irrepresentability does not hold. -\begin{comment} \begin{lemma} Let ${\cal C}({\cal M}, \bar {\cal M}^\perp, \theta^*) \defeq \{ \Delta \in \mathbb{R}^p | {\cal R}(\Delta_{\bar {\cal M}^\perp} \leq 3 {\cal R}(\Delta_{\bar {\cal M}} + 4 {\cal R}(\theta^*_{{\cal M}^\perp}) \}$, where $\cal R$ is a \emph{decomposable} regularizer with respect to $({\cal M}, \bar {\cal M}^\perp)$, and $({\cal M}, \bar {\cal M})$ are two subspaces such that ${\cal M} \subseteq \bar {\cal M}$. Suppose that $\exists \kappa_{\cal L} > 0, \; \exists \tau_{\cal L}, \; \forall \Delta \in {\cal C}, \; {\cal L}(\theta^* + \Delta) - {\cal L}(\theta^*) - \langle \Delta {\cal L}(\theta^*), \Delta \rangle \geq \kappa_{\cal L} \|\Delta\|^2 - \tau_{\cal L}^2(\theta^*)$. Let $\Psi({\cal M}) \defeq \sup_{u \in {\cal M} \backslash \{0\}} \frac{{\cal R}(u)}{\|u\|}$. Finally suppose that $\lambda \geq 2 {\cal R}(\nabla {\cal L}(\theta^*))$, where ${\cal R}^*$ is the conjugate of ${\cal R}$. Then: $$\|\hat \theta_\lambda - \theta^* \|^2 \leq 9 \frac{\lambda^2}{\kappa_{\cal L}}\Psi^2(\bar {\cal M}) + \frac{\lambda}{\kappa_{\cal L}}\{2 \tau^2_{\cal L}(\theta^*) + 4 {\cal R}(\theta^*_{{\cal M}^\perp}\}$$ \end{lemma} |